Source
Debreu (1952), “A Social Equilibrium Existence Theorem”, PNAS 38(10):886–893, LIT-067, read in full in NOTE-037 from an archival scan, with the theorem, lemma, proof and Remark checked against page images.
What was actually shown
The setting. ν agents. Agent ι chooses a_ι from A_ι(ā_ι), a subset of its action set 𝔄_ι that depends on the others’ actions ā_ι, and maximises f_ι. An equilibrium is a profile in which each a_ι is feasible given ā_ι and maximises f_ι there (Definition, p. 888). This is now called an abstract economy or generalized game.
The theorem (p. 888). An equilibrium exists if, for every ι:
- 𝔄_ι ⊂ ℝ^{n_ι} is a contractible polyhedron;
- A_ι has non-empty compact values and a closed graph G_ι;
- f_ι is continuous on G_ι into the extended reals;
- the value function φ_ι(ā_ι) = max f_ι over A_ι(ā_ι) is continuous;
- each best-response set M_ā_ι is contractible.
The proof (p. 889). The product of the 𝔄_ι is a contractible polyhedron. The map sending a profile to the product of best-response sets has contractible values and a closed graph. A Lemma, a case of Eilenberg–Montgomery (1946) or Begle (1950), gives such a map a fixed point, and a fixed point is an equilibrium.
The Remark (p. 889). If G_ι is compact, A_ι is continuous in Debreu’s sense (lower hemicontinuous, in modern terms) and f_ι is continuous, then φ_ι is continuous. Debreu flags the continuity of φ_ι as “a joint requirement … not well adapted to applications”, which the Remark answers.
The saddle-point Corollary (§3). With contractible argmin and argmax sets, a continuous f on a product of contractible polyhedra has a saddle point. This contains the theorems of Kakutani and of von Neumann, since convex sets are contractible.
What could have come out otherwise. Existence could have needed convex best-response sets, as in Kakutani and Nash. It needs only that they be topologically trivial: “The convexity assumptions were, however, irrelevant” (§4, p. 892).
What this does not say
- Uniqueness, stability or a way to compute an equilibrium. The proof is a fixed-point argument and gives no algorithm or rate.
- That the theorem is shown to contain Nash’s. The introduction asserts it, and §4 only restates Nash’s theorem. The reduction (constant constraint equal to a mixed-strategy simplex, convex best responses, the Remark for continuity) is routine but left to the reader (NOTE-037, C4).
- That the hypotheses are easy to check. Contractibility of best-response sets is hard to verify outside convex or quasi-concave settings.
- That it covers infinite dimensions, infinitely many agents or incomplete preferences. Payoffs are real-valued, so preferences are complete orderings. Later work weakens these; it is not assessed here.
- That every agent is a contractible polyhedron. The conditions are on action sets and best-response sets, not on agents.
Connections
- The record’s other game-theory accounts concern evolutionary and repeated games. This is the existence theorem under Arrow and Debreu’s competitive equilibrium.